The Beauty of Symmetry: An Introduction to the Wallpaper Group
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- Опубликовано: 10 июн 2024
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Resources used in the video:
Web app to generate all 17 wallpapers: math.hws.edu/eck/js/symmetry/...
Lots of images for each element of the wallpaper group: en.wikipedia.org/wiki/Wallpap...
The aperiodic monotile: cs.uwaterloo.ca/~csk/hat/
There are precisely 17 different periodic tilings of the plane. That is, demanding translational symmetries, we can investigate the possible rotational, mirror reflection and glide reflection symmetries and it creates the so called Wallpaper group with 17 possibilities.
0:00 The wallpapers of Alhambra
1:07 The four symmetries
2:31 The symmetries of pmg
5:55 Tilings and lattices
8:02 the 17 wallpapers
10:20 The symmetries of p6
12:45 The symmetries of cm
13:50 Plotting all 17 wallpapers
14:28 Why only rotations of 2,3,4,6
17:02 New research on aperiodic tilings
17:50 brilliant.org/TreforBazett
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I think your cell for P2 (at 10:10) is too big. From my understanding, the corners should be the 2 red and 2 yellow rhombuses. The yellow and the red are equal, and the green and blue are also equal. Between the red and yellow rhombuses, there are two-fold rotation axis you haven't showed. As well as between green and blue.
Oh, good catch, you're exactly right. I sometimes forget that p2 is actually compatible with all of the underlying lattices, and so you can look for rectangles as the primitive cell not just a square like I've done.
@@DrTrefor Thank you. I wondered why the pattern around the yellow and red rhombuses look identical but have different colors. Because of this, I looked closer and found the 2 missing axes of rotation. Are you going to make a follow-up video about space groups or point groups?
@@DrTrefor Sir can u mention any good book for beginners to this type of tilings and wallpapers problem including periodic and aperiodic tilings?
He doubled the unit cell in the horizontal direction
Wonderful video! Little nitpick at the end: the important property of this new monotile is that it can **only** tile the plane aperiodically. There already were single tiles that can tile the plane aperiodically, but they also can periodically. For example, 45/45/90 triangles can tile the plane aperiodically kind of trivially just by forming squares with the hypotenuse / or \ randomized.
The extension of this concept to 3D leads to 230 space groups. This is a really important concept on the study materials behavior/properties. It would be great have a similar video to the 3d case.
Ya I think a bunch of material science engineering depends on these ideas
never been this early to a video, so let me say -- thank you for providing so much awesome, free content Trefor!
Thank you so much - appreciate that!
5 possible lattices and 17 tilings on a flat 2D surface. all prime numbers.
the proof must be so satisfying
Great video! I like that you showed the different symmetries using real artistic examples.
I found out about this many years ago because I looked up interesting properties of the number 17 (as one does), but this is definitely a topic that deserves more attention!
Glad you enjoyed!
One of the early diskworld books (Terry Pratchett) has a temple with octagonal tiles, yes it was deliberate, yes I suspect many readers missed the impossibility and, yes under those circumstances you should run for the exit.
Oh nice! Right you can combine octagonal tiles and square tiles but not just the former
I'd love to see someone make ceramic tiles of the hat so you can aperiodically tile your backsplash.
That would be amazing!
Great presentation! Thank You!
Reminds me of my introduction to algebra textbook, good times.
This is a very interesting video! thank you
Amazing content!!
Thanks for the video! Penrose tilings always seemed a bit counterintuitive, but fascinating.
Beautiful!
Loved the video. Envious of the T-shirt
Ha thanks! It's an awesome one (also check out merch link in description:) )
Brilliant.
Please do more videos on mathematical art
Given a wallpaper, how can we draw the unit cell? at first glance it seems there are multiple possibilities on how the pattern repeats
You can always make shifts or you can zoom out and get cells that are bigger than the minimum
If you want to take advantage of the other symmetries, you can make the boxes even smaller. If you treat color swaps as a kind of symmetry, then the entire first pattern can be generated from just a single small right-triangle.
totally unrelated but where did you get that shirt: sleep, eat, math, repeat with the yellow square-circles from?
What???? Looks interesting 🤔
This must be interesting as all your vids are interesting!
Thanks for saying!
@@DrTrefor who is liking us all?
What happens if you relax the requirement that you must be able to _translate_ every point to any other point? With the 6-fold flower pattern, you said that the two 3-fold symmetries were different, and yet using the 2-fold rotation between them, they each turn into the other. Similarly, the 2-fold rotations you said were different can be transformed into each other with the 3-fold rotations. There were also a few tilings that had _color_ symmetry, where a smaller translation than normal _combined with a color swap_ leaves the pattern unchanged. Applying this would combine the two 2-fold rotations for the first tiling to just one.
While I can accept that 5-fold rotation is impossible on a lattice, it clearly _isn't_ impossible for a more general tiling since that's exactly what the Penrose Tiling has, even if only at a single point.
G. Pólya wrote in preface of his book *How to Solve it* : Mathematics has two faces, it is the rigorous science of Euclid but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but *Mathematics in the making appears as an experimental inductive science* . Both the aspects are as old as the science of Mathematics itself. But the second aspect is new in one respect, *Mathematics in the process of being invented, has never before been presented in quite this manner to the student, or to the general public* .
But sometimes it may not be possible to reinvent the proof by ourselves, as the proof may be long, logically complicated and rely on some really clever insight. This tends to worry us. Okay, we can see how the proof work, but *we would never have thought of that* . And sometimes it make us wonder whether we are good enough at Mathematics.
So my question is in general when we encounter such a proof How to interact with them ? How to make best out of such a proof ?
Sir please give us some nice advice...
Leibniz talks about precisely this point in his amazing lesser known work New Essays Concering Human understanding!
While many things we know a priori independently of experience (math was by far the most paradigmatic and uncontroversial example of this, so much so that Hume, the 'arch empiricist' granted math's a priori status). But sometimes it requires sense experience to awaken these intuitions and even contribute to their status as knowledge (but never with the same certainty as a deductive proof).
the world of aperiodic tiling sounds pretty wild... can it be thought of as random? how many types are there?
i wonder if there is a pattern on the density of shape (how many units of a given shape can be found on a given area of aperiodic tile), even though there is no repetition
Here is an example of aperiodic, nonrandom tiling, built with regular pentagons and 36⁰ acute-angles rhombuses the same side as the said pentagons:
Surround a central pentagon with five other pentagons attached to it side by side. Place rhombuses between them. Then repeat, surrounding such a disk with other pentagons and placing rhombuses between them to fill gaps again. Repeat the extending procedure ad nauseam. The radial symmetry and the order-5 mirror symmetries are conserved at each iteration.
@@wafikiri_ but no translation symmetry? is that why its aperiodic?
@@GeoffryGifari Exactly.
oh oh... i thought that, if these patterns are the only ones possible to tile a flat wall, what about curved surfaces? how many regular patterns can cover a spherical surface? maybe you guys know
That's a great question!
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I must disagree with that number, 17. Because I am able to provide an infinite number of different periodic tilings. A series of tilings equivalent to the natural numbers. It was not easy, but not too difficult, either.
One of those tilings has five-fold rotational symmetry and ten mirror symmetries but no translational symmetry. All the other tilings discovered by me have just translational symmetry, generally in only one direction, although some have it in more.
The claim about the 17 isn't that there isn't infinitely many possible tilings, but that there are only 17 possible sets of symmetries of those tilings under the rules suggested in the video. So for example, none of your infinite tilings presumably have 5 fold rotational symmetries.
@@DrTrefor Rotational symmetry is not required in wallpapers. Translational periodic symmetry is what all the tilings in the series I mentioned have, and an infinite subset of them have two-fold rotational symmetry. Also an infinite subset of them have translational symmetry in two directions. But the tilings do not fit in rectangles, triangles, hexagons, etc., because they are composed of pentagons and rhombuses: the lattices they form, though periodic, are more complex. They all can be inscribed (at least theoretically: most would require extremely long cylinders to preserve their individuality, the same way most numbers have long lengths when expressed in the decimal system) on a printing cylinder to print wallpapers.
@@DrTrefor Now I understand that you mean that the infinite series of infinite periodic tilings can be classified in a subset of those 17 symmetry cases. Sorry for being slow.
first
Nice!
Tiring to listen to the over the top enunciation…
Better than monotone